In this study, we examine the statistical properties of asymmetric surface dielectric barrier discharges (SDBD) produced by applying a periodic high voltage between two conducting displaced electrodes, located at the opposite sides of a flat dielectric panel. Here, the asymmetry refers to the fact that the lower electrode is fully covered with an insulating material, while the upper one, glued onto the dielectric surface, is otherwise left exposed to the air. Such a configuration allows the formation of a thin layer of plasma above the insulating surface. A single cycle signal consists of two well-separated half-cycle patterns, denoted as forward and backward strokes, corresponding to positive and negative voltages, respectively. They display a quite complex discharge pattern constituted by a sequence of individual peaks (bursts) of varying current and time duration. Specifically, we find that backward stroke bursts carry a positive mean charge Q≃0.3 nC and mean current I≃35 mA, with a mean duration τ≃15 ns, while forward stroke bursts have a negative mean charge Q≃−0.1 nC, a mean current I≃−20 mA, and a mean duration τ≃11 ns. The statistical analysis suggests that power injection can be tailored to produce the active agents in the plasma needed for a particular application. We also determined discharge spatial correlation patterns from measurements of the associated stimulated optical emission. The optical excitations occur as a result of the ionizing effect of the electromagnetic waves which ignite the discharge, followed by the electric current flow. In particular, we point out that one of the phases of the discharge is compatible with a cathode directed streamer phenomenon (backward stroke), while the mechanism acting for a forward stroke has a different structure.